Integral and series representations of the digamma and polygamma functions

Details Integral and series representations of the digamma and polygamma functions Integral and series representations of the digamma and polygamma functions

2010-07-31

arxiv.org/abs/1008.0040v2

Physics

Mathematical Physics

27 pages, no figures

Scientific paper

We obtain a variety of series and integral representations of the digamma function $\psi(a)$. These in turn provide representations of the evaluations $\psi(p/q)$ at rational argument and for the polygamma function $\psi^{(j)}$. The approach is through a limit definition of the zeroth Stieltjes constant $\gamma_0(a)=-\psi(a)$. Several other results are obtained, including product representations for $\exp[\gamma_0(a)]$ and for the Gamma function $\Gamma(a)$. In addition, we present series representations in terms of trigonometric integrals Ci and Si for $\psi(a)$ and the Euler constant $\gamma=-\psi(1)$.

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